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classes of numerical schemes have been developed for classical mechanics ( Leimkuhler and Reich 2004 ; Hairer et al. 2006 ). Of particular interest in this field are conservation laws for energy, symmetry-induced conservation laws, and preservation of the symplectic (i.e., intrinsic geometric) structure. These methods, primarily designed for finite-dimensional systems of ordinary differential equations, are now commonly referred to as geometric integration. Extension of geometric integration to
classes of numerical schemes have been developed for classical mechanics ( Leimkuhler and Reich 2004 ; Hairer et al. 2006 ). Of particular interest in this field are conservation laws for energy, symmetry-induced conservation laws, and preservation of the symplectic (i.e., intrinsic geometric) structure. These methods, primarily designed for finite-dimensional systems of ordinary differential equations, are now commonly referred to as geometric integration. Extension of geometric integration to
equation evolution has an exact material invariant, the PV, for frictionless, adiabatic flow—accurately invertible in a surprising range of circumstances—it might be useful to make the balanced model represent PV evolution as accurately as possible. This leads to the idea of trying to incorporate exact material PV conservation from the start, by formulating the model in such a way that its velocity field v , obtained by PV inversion, both advects and evaluates the PV (e.g., Bleck 1974 ; Norton 1988
equation evolution has an exact material invariant, the PV, for frictionless, adiabatic flow—accurately invertible in a surprising range of circumstances—it might be useful to make the balanced model represent PV evolution as accurately as possible. This leads to the idea of trying to incorporate exact material PV conservation from the start, by formulating the model in such a way that its velocity field v , obtained by PV inversion, both advects and evaluates the PV (e.g., Bleck 1974 ; Norton 1988
equations) only for the case of uniform shear, though other background wind profiles are sometimes found in the literature. For this case, the 2DPE have been shown to conserve total energy, as well as potential vorticity on parcels (M. D. MacKay and G. W. K. Moore 1996, unpublished manuscript). In this brief article the uniform shear 2DPE model is shown to be Hamiltonian, and a symplectic representation, Casimir invariants, and a pseudoenergy conservation law are all found. 2. Mathematical model The
equations) only for the case of uniform shear, though other background wind profiles are sometimes found in the literature. For this case, the 2DPE have been shown to conserve total energy, as well as potential vorticity on parcels (M. D. MacKay and G. W. K. Moore 1996, unpublished manuscript). In this brief article the uniform shear 2DPE model is shown to be Hamiltonian, and a symplectic representation, Casimir invariants, and a pseudoenergy conservation law are all found. 2. Mathematical model The
1. Introduction The conservation equations of oceanography are treated in many texts, including Batchelor (1967) and Gill (1982) . In theoretical analysis and numerical modeling, it is traditional to make the Boussinesq approximation where every appearance of density is replaced by a fixed reference density except in the buoyant force in the vertical momentum equation. Since the reference density is usually taken to be 1000 kg m −3 and since the in situ density of seawater can be as large
1. Introduction The conservation equations of oceanography are treated in many texts, including Batchelor (1967) and Gill (1982) . In theoretical analysis and numerical modeling, it is traditional to make the Boussinesq approximation where every appearance of density is replaced by a fixed reference density except in the buoyant force in the vertical momentum equation. Since the reference density is usually taken to be 1000 kg m −3 and since the in situ density of seawater can be as large
1. Introduction Ideal fluid motion obeys many conservation laws (mass, energy, linear–angular momentum, potential vorticity). Certain forms of the equations of motion expose explicitly these conservation properties. Such forms are a good starting point for devising numerical methods that inherit from the continuous equations as many of their conservation properties as possible. For example, the flux form of the equations is the basis of finite-volume methods, which inherently conserve mass
1. Introduction Ideal fluid motion obeys many conservation laws (mass, energy, linear–angular momentum, potential vorticity). Certain forms of the equations of motion expose explicitly these conservation properties. Such forms are a good starting point for devising numerical methods that inherit from the continuous equations as many of their conservation properties as possible. For example, the flux form of the equations is the basis of finite-volume methods, which inherently conserve mass
VOL. 49, NO.I JOURNAL OF THE ATMOSPHERIC SCIENCES 1JANUARY1992Nonlinear Wave-Activity Conservation Laws and Hamiltonian Structure for the Two-Dimensional Anelastic Equations J. F. SCINOCCA AND T. G. SHEPHERD Department of Physics, University of Toronto, Toronto, Canada(Manuscript received 10 January 1991, in final form 15 May 1991)ABSTRACT Exact, finite-amplitude, local wave-activity conservation laws are derived for ff~sturbanees to
VOL. 49, NO.I JOURNAL OF THE ATMOSPHERIC SCIENCES 1JANUARY1992Nonlinear Wave-Activity Conservation Laws and Hamiltonian Structure for the Two-Dimensional Anelastic Equations J. F. SCINOCCA AND T. G. SHEPHERD Department of Physics, University of Toronto, Toronto, Canada(Manuscript received 10 January 1991, in final form 15 May 1991)ABSTRACT Exact, finite-amplitude, local wave-activity conservation laws are derived for ff~sturbanees to
species of a droplet. The soluble species is absorbed into the liquid, and other K − 2 components do not undergo phase transition at the droplet surface. The droplet is heated by a conduction heat flux from the high temperature surroundings and begins to evaporate. In the further analysis we assume spherical symmetry and neglect effects of buoyancy and thermal diffusion. Under these assumptions, the spherically symmetric system of mass and energy conservation equations for the liquid phase 0 < r
species of a droplet. The soluble species is absorbed into the liquid, and other K − 2 components do not undergo phase transition at the droplet surface. The droplet is heated by a conduction heat flux from the high temperature surroundings and begins to evaporate. In the further analysis we assume spherical symmetry and neglect effects of buoyancy and thermal diffusion. Under these assumptions, the spherically symmetric system of mass and energy conservation equations for the liquid phase 0 < r
background state. If the background is defined using the Eulerian zonal mean of the full flow, as in Charney and Stern (1961) , the global conservation law is not respected exactly at large amplitude. However, McIntyre and Shepherd (1987) formulated a general recipe to construct Eulerian measures of wave activity that are conserved exactly at large amplitude when measured relative to a zonally symmetric background state that is a solution of the governing fluid equations. It is possible to specify a
background state. If the background is defined using the Eulerian zonal mean of the full flow, as in Charney and Stern (1961) , the global conservation law is not respected exactly at large amplitude. However, McIntyre and Shepherd (1987) formulated a general recipe to construct Eulerian measures of wave activity that are conserved exactly at large amplitude when measured relative to a zonally symmetric background state that is a solution of the governing fluid equations. It is possible to specify a
DECEMBER 1996 MCDOUGALL AND MCINTOSH 2653The Temporal-Residual-Mean Velocity. Par~ ]I: Derivation and the Scalar Conservation Equations TREVOR J. McDOUGALL AND PETER C. MCINTOSHAntarctic CRC, University of Tasmania, and CSIRO Division of Oceanography, Hobart, Tasmania, Australia(Manuscript received 17 January 1996, in final form 7 May 1996)ABSTRACT The time
DECEMBER 1996 MCDOUGALL AND MCINTOSH 2653The Temporal-Residual-Mean Velocity. Par~ ]I: Derivation and the Scalar Conservation Equations TREVOR J. McDOUGALL AND PETER C. MCINTOSHAntarctic CRC, University of Tasmania, and CSIRO Division of Oceanography, Hobart, Tasmania, Australia(Manuscript received 17 January 1996, in final form 7 May 1996)ABSTRACT The time
any lead time. Many studies in which singular vectors have been derived by direct linearization of the nonlinear governing equations fail to be definitive either because they make unrealistic assumptions about the initial condition, or use a linear model that is not derived from the nonlinear system that generated the initial conditions. For example, Sardeshmukh et al. (1997) considered an ensemble of initial conditions drawn from real observations but used a barotropic model to predict them
any lead time. Many studies in which singular vectors have been derived by direct linearization of the nonlinear governing equations fail to be definitive either because they make unrealistic assumptions about the initial condition, or use a linear model that is not derived from the nonlinear system that generated the initial conditions. For example, Sardeshmukh et al. (1997) considered an ensemble of initial conditions drawn from real observations but used a barotropic model to predict them
